Measuring Inequality from Top to Bottom

Transcription

1 Policy Research Working Paper 7237 WPS7237 Measuring Inequality from Top to Bottom Tania Diaz-Bazan Pulic Disclosure Authorized Pulic Disclosure Authorized Pulic Disclosure Authorized Pulic Disclosure Authorized Poverty Gloal Practice Group April 215

2 Policy Research Working Paper 7237 Astract This paper presents a new methodology to measure inequality that optimally comines household survey information and tax records to construct a complete income distriution. Comining the two data sources is necessary ecause, on the one hand, household surveys do not accurately represent the wealthiest segment of the population, while tax records do; on the other hand, the opposite is true for the lower end of the income distriution: tax records only include incomes aove a certain threshold. The key innovation of the proposed methodology and the main difference from the existing literature is the choice of an optimal income threshold. The Gini coefficient for the population is then computed comining the conditional income distriutions for incomes elow (using household survey data) and aove (using tax records). Central to this methodology is the fact that is not chosen aritrarily: it should e determined in such a way as to minimize reliance on household survey data to compute the top of the income distriution. In practice, the optimal corresponds to the minimum income level that triggers mandatory tax filing. The proposed methodology is applied to the case of Colomia. This paper is a product of the Poverty Gloal Practice Group. It is part of a larger effort y the World Bank to provide open access to its research and make a contriution to development policy discussions around the world. Policy Research Working Papers are also posted on the We at The authors may e contacted at tdiazazan@ worldank.org or diaztania@gmail.com. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas aout development issues. An ojective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should e cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced y the Research Support Team

3 Measuring Inequality from Top to Bottom Tania Diaz-Bazan World Bank I am grateful to my thesis advisor Alerto Porto for early discussions on the topic, and for his constant and helpful supervision in developing the idea. I am also grateful to Stefano Giglio for his thoughtful suggestions and to Louise Cord, Lea Gimenez Duarte, Carlos Rodriguez Castelan, and Wendy Cunningham for many helpful discussions. tdiazazan@worldank.org and diaztania@gmail.com.

4 1 Introduction Welfare measures such as poverty and inequality are central to the study of economic development. The causes and consequences of inequality have always een a key concern for governments and scholars, ut interest in the evolution of inequality has recently spiked. This paper focuses on how to optimally measure inequality. Household surveys have traditionally een the main data source to estimate welfare measures like poverty and inequality. However, they are suject to measurement error, underreporting, and non-response, among other prolems (Székely and Hilgert 1999, Cowell and Victoria-Feser 23, Atkinson 27, Burkhauser et al. 29, Burkhauser et al. 212). These issues represent a particularly severe prolem when measuring inequality: mostly affect the top of the income distriution, and this can lead to an incorrect estimation of the levels and trends in the Gini coefficient. If non-response rates were constant over time, the measurement of changes in inequality over time would not e greatly affected y these prolems (Gasparini et al. 2). However, it is likely that these rates do fluctuate over time. 1 they The income underreporting level at the top of the distriution is not constant and homogeneous it can vary y economic strata, y source of income, and over time. Yet there is no mechanism availale to detect and model precisely the level of underreporting in household surveys. The most notale pattern is that richer households are more hesitant to disclose their income and assets, which would explain why underreporting tends to affect the top of the income distriution, ultimately resulting in an underestimation of the Gini coefficient (Székely and Hilgert 1999). Time variation in underreporting at the top of the income distriution will also affect the estimated time series of the coefficients (see Burkhauser et al. 24, 29, Piketty and Saez 26, and Atkinson 27). Finally, household surveys may fail to represent all types of income sources (for example capital gains), which may result in further ias (Burkhauser et al. 212). An alternative data source used to study top incomes in a population is income tax 1 Statistical institutes have partly succeeded in addressing the non-response issue in household surveys through the use of multivariate regressions with demographic and socioeconomic characteristics. 1

5 records, which contain extensive information on firms and individuals that are often not captured y traditional data collection mechanisms such as household surveys. Therefore, tax records can provide valuale input to improve the measurement of income inequality, with the caveat that they also present important limitations. Not only do tax records contain little information aout the ottom segments of the income distriution (where individuals are not required to file a tax return), ut also filers have the financial incentive to report their income in a way that limits their tax liailities, therefore reducing the marginal tax rate (Burkhauser et al. 212), and the estimation could e iased due to tax evasion. That said, examining the top of the income distriution is essential to understanding inequality, and tax records are a fundamental source for this purpose. As shown y the key contriutions of Atkinson (27) and Alvaredo (211), the share of income concentrated at the top of the income and wealth distriution can have dramatic effects on inequality. Recent studies have therefore proposed to rely on oth household surveys and tax records to improve the estimation of the Gini coefficient. In particular, Atkinson (27) proposes an approximate methodology to estimate the Gini coefficient. The first stage of his methodology consists in choosing a group of top income earners (say, the top.1 percent). Then, if this group of top earners is approximately an infinitesimal fraction of the population, with share S of total income, the Gini coefficient can e approximated y the following calculation: S + (1 S )G. In this formula, G is the Gini coefficient of the population that excludes the top earners. More recently, Alvaredo (211) extends this methodology to the case in which the group of top earners chosen is not infinitesimal relative to the size of the population (for example, the top 1 or 5 percent of the income distriution). These methodologies have two salient features worth noting. First, the choice of which group of top earners to consider in the formula is aritrary the top income group is not optimally chosen ased on specific criteria. Second, the Gini coefficient G for the population that does not include the chosen top earners group is computed using the household surveys. This means that, since we cannot determine precisely the extent of underreporting at the top of the income distriution, it is not clear that the G estimated from household surveys 2

6 will correctly capture the Gini coefficient of the population that excludes the selected top earners. For example, suppose that we choose to apply the methodology focusing on the top.1 percent of the population, so S in the formula aove would e the share of income of the top.1 percent computed from tax records. Suppose also that the household survey is only representative of the ottom 95 percent of the population, ecause individuals aove the 95th percentile do not correctly report their income. In that case, the G computed from the household survey would not correspond to the true Gini coefficient of the population outside the selected group of top earners (in this case the ottom 99.9 percent), since the percentiles are not covered at all in the household survey. The methodology would then suffer from omitting a nontrivial segment of the population from the analysis. In this paper we uild upon the existing methodologies (particularly, Atkinson 27 and Alvaredo 211) and propose a new methodology aimed to optimally comine the two data sources, accounting explicitly for the underreporting of the top of the income distriution in the household survey. The methodology is ased on finding the optimal income threshold such that the following statements hold true: (1) the segment of the population with incomes elow is well represented in the household survey data; (2) those with incomes aove are captured in the tax records; and (3) minimizes the distortions attriuted to household survey underreporting at the top of the income distriution. This paper shows that if can e found (possily elow the income level of the top.1 or 1 percent), the population s full unconditional distriution of income can e estimated y comining the Gini coefficients of the population segments with incomes elow (using household survey data) and aove (using tax records). Since the underreporting distortions in the household survey are stronger for higher incomes, the optimal that satisfies these requirements is the lowest level of income that triggers mandatory tax filing. This is the key element of the proposed methodology. Note that this income threshold can e significantly elow the 1 percent traditionally used in the existing literature. The proposed methodology, therefore, solves the issue of aritrarily choosing the top income group, instead relying on the optimal income threshold, which gives the maximum weight possile to tax records at the top of the income distriution. In addition, the 3

7 methodology explicitly takes into account the underreporting of household surveys at the top of the income distriution, and only relies on household survey data to compute the Gini coefficient for the segment of the income distriution that lies elow the threshold. This paper is organized as follows. After presenting the methodology and its properties in Section 2, the paper demonstrates an application to the case of Colomia (Section 3). Although in recent years an increasing numer of governments have made income tax data availale to the pulic, Latin American countries have lagged ehind. Colomia is the first country in the region to make personal income tax micro-data availale to researchers, which determined the case study selection in this analysis. Section 4 offers conclusions. 2 Methodology This section descries a simple procedure to estimate the entire income distriution in a population y comining the information in two data sets that are only informative aout conditional income distriutions. In this case, we focus on the fact that household surveys are representative of the ottom segment of the income distriution ut not the top, while tax records capture the top end of the income distriution ut not the ottom. The proposed methodology applies a result of Dagum (1997), who showed how to decompose a population s Gini coefficient into a comination of the Gini coefficients of its supopulations. A related methodology has een applied y Alvaredo (211) to measure income inequality in the United States and Argentina. The end of this section summarizes the main differences etween the proposed methodology and the previous literature. 2.1 Setup and Assumptions Suppose that the cumulative distriution function of income y in a population is F (y). We would like to otain an estimate of F (y) and some related statistics (like the Gini coefficient, G[F ]). However, we can only otain consistent estimates of some conditional distriutions of y. In particular, we assume: 4

8 Assumption 1. Define F 1 (y) = P r{y y Y } F 2 (y) = P r{y y Y > } for some. We oserve ˆF 1 (y) and ˆF 2 (y), pointwise consistent estimators of F 1 (y) and F 2 (y) respectively. The first assumption simply states that out of the availale information (for example, household surveys or tax records) we can otain consistent estimators of the conditional distriution of income; ˆF 1 estimates the conditional distriution elow an income threshold, while ˆF 2 estimates the conditional distriution for incomes aove. Assumption 2. ˆF (), a consistent estimator of F (), is oserved. The second assumption is necessary to comine the information in ˆF 1 and ˆF 2 correctly into an estimator of F. This assumption does not require knowing the full distriution F, only the value at point. 2.2 Results We provide two simple results. The first proposition shows that under assumptions 1 and 2, we can otain a consistent estimate of the distriution F. The second proposition shows that it is possile to express the Gini coefficient of the income distriution F as a comination of Gini coefficients constructed using the conditional distriutions F 1 and F 2. Proposition 1. Under Assumptions 1 and 2, the estimator ˆF (y) constructed as: ˆF (y) ˆF () ˆF 1 (y) ˆF () + (1 ˆF ()) ˆF 2 (y) y y > is a pointwise consistent estimator of F (y). 5

9 Proof. In the population, y the definition of conditional distriution we have, for y, F 1 (y) P r{y y Y } = P r{y y} P r{y } = F (y) F () so F (y) = F 1 (y)f () and for y > F 2 (y) P r{y y Y > } = P r{y y and Y > } 1 P r{y } = F (y) F () 1 F () so F (y) = F () + (1 F ())F 2 (y) Given that under Assumptions 1 and 2 we have consistent estimators for all the right-hand side variales, the consistency of F (y) follows immediately. Proposition 1 tells us how we can reconstruct the underlying unconditional income distriution if we know the two conditional distriutions aove and elow an income threshold and the fraction of the population with income lower than. From the estimated ˆF (y), we can then study the properties of the income distriution. In some cases, it may e convenient to e ale to otain directly the Gini coefficient for the full population, G [F ]. The next proposition shows that the Gini coefficient G[F ] can e otained as a simple linear comination of the Gini coefficients computed on the conditional distriutions. We start from a few definitions. Definition. The Gini coefficient G[F ] is defined as: G[F ] = 1 F (y)(1 F (y))dy where E[Y ] is the unconditional mean of Y (i.e., the population mean). 6

10 Definition. The Gini coefficient of a conditional distriution G[F j ], j {1, 2}, is defined as: G[F j ] = 1 F j (y)(1 F j (y))dy j where 1 E[Y Y ] and 2 E[Y Y > ]. 2 These Gini coefficients are simply the standard Gini coefficients computed on each of the two distriutions. Proposition 2. The full-population (unconditional) Gini coefficient can e written as: where G[F ] = (1 F ())F ()[ ( 2 1 ] + F () 2 ) ( 1 G[F 1 ] + [1 F ()] 2 ) 2 G[F 2 ] Proof. See Appendix. = F () 1 + (1 F ()) 2 This proposition tells us that we can estimate the overall Gini coefficient as a linear comination of the Gini coefficients of the conditional distriutions, plus an adjustment term that takes into account that the two conditional distriutions are otained from different underlying portions of the unconditional distriution. 2.3 Implementation and Relation to the Literature A crucial prerequisite in the derivation of Propositions 1 and 2 is that we can otain a consistent estimate of the conditional distriution of income aove and elow a threshold. The specific case of comining household surveys with tax records provides a natural example in which to apply this methodology. Household surveys tend to e poorly representative of the top of the income distriution ecause high income individuals tend to underreport or simply not report their 2 Note that given the definitions of F 1 and F 2 aove, F 1 (y) is 1 for y >, so F 1 (y)(1 F 1 (y))dy = F 1(y)(1 F 1 (y))dy. Viceversa, F 2 (y) is for y, so F 2 (y)(1 F 2 (y))dy = F 2 (y)(1 F 2 (y))dy. 7

11 income. While the survey data will have a ias that primarily affects the top of the income distriution, the ottom segment of the population might e much etter represented in the survey. For simplicity purposes, suppose that all incomes elow a threshold a are well captured in the household survey. Similarly, for the case of tax records, only individuals with incomes aove a threshold c report taxes, so only this high-income population is well represented. In this case, the dataset will correctly represent the conditional distriution of income aove that threshold. Suppose now that c < a. In other words, suppose that the level of income that triggers tax reporting is elow the level of income aove which we fear significant censoring in the household survey. Then, any income level etween c and a (c a) will satisfy the following two properties: (1) all incomes aove are well represented in the tax records data (since it is aove the minimum threshold c that triggers tax filing); and (2) all incomes elow are well represented in the household survey, ecause censoring only occurs aove a, and < a. Any such would e a valid income threshold to apply the methodology aove. In practice, it is hard to know where censoring in the household survey starts (a). This suggests that rather than viewing as a purely free parameter, one should optimally choose to minimize the potential ias due to household survey underreporting at the top of the income distriution. If tax records are representative of incomes aove the minimum threshold to file (c), the natural choice for the optimal would e precisely c. In other words, since we fear that household surveys underreport at the top of the income distriution, we optimally choose the lowest possile that still ensures that incomes aove are represented in the tax records. The natural choice is the income level that triggers mandatory tax filing. This intuition aout the optimal threshold and its implications for the estimation of inequality measures are the distinguishing features of the methodology presented in this paper relative to the existing literature. In particular, the methodology of Alvaredo (211) applied to the case of Colomia y Alvaredo and Londoño (213) comines an estimate of the Gini coefficient for the ottom 99 percent of the income distriution (G ) with 8

12 the share of income of the top 1 percent (S), the fraction of the population (P ) in that group (P = 1%), and an estimated coefficient of the inverted Pareto distriution (β) that captures the shape of the income distriution function within the 1 percent. The formula used is: G = (β 1)/(β + 1)P S + G (1 P )(1 S) + S P ; G is estimated using the household survey, while S and β are otained from tax records. In this paper, we uild upon the contriution of Alvaredo (211) and propose a methodology that differs from those in several dimensions. First, their formula otains an estimate of the Gini coefficient of the ottom 99 percent of the population (G ) from the household survey. However, as previously noted, household surveys are not representative of the full population, which means that G does not capture the true ottom 99 percent of the population. Estimating G y computing the Gini coefficient of the ottom 99 percent of those households that are present in the household survey will yield a iased result if there is survey underreporting at the top of the income distriution (since it will use the ottom 99 percent of a sample which is truncated at the top, not of the true income distriution). Alternatively, G can e estimated as the Gini coefficient of the entire household survey (without excluding the top 1 percent of the respondents), under the assumption that the household survey is effectively representative of the ottom 99 percent of the income distriution due to underreporting at the top. This approach would e correct if we were certain that the household survey misses precisely the top 1 percent. However, if household survey underreporting is significant at a different income level (for example, elow the top 1 percent), the estimated G would not correspond to the Gini coefficient of the ottom 99 percent of the true income distriution. A second difference etween the two approaches is that, even assuming an accurate estimation of G, the formula only corresponds (up to the approximation discussed in Alvaredo 211) to the one presented in this paper with a choice of equal to the 99th income percentile, as opposed to the optimal discussed aove. As previously argued, we elieve that choosing a lower (corresponding to the income threshold that triggers mandatory tax filings) results in a etter estimate ecause it reduces the reliance on household survey data for the top of the income distriution. 9

13 As shown empirically in Section 3, oth features result in interesting differences etween the results of Alvaredo and Londoño (213), ased on Alvaredo (211), and the ones we otain in this paper for the same country, Colomia. F () that appears in the equation of Propositions 1 and 2. A final note concerns the term This term represents the fraction of the population with incomes elow, and needs to e estimated once the preferred threshold is chosen. We estimate F () using the same methodology employed y Atkinson (27) and Alvaredo (211) to estimate the share of income that accrues to the top 1 percent of the income distriution. In particular, F () = 1 A, where A is the C total population aove the income tax threshold (computed using the tax records), while C is the control for population (an estimate of the total population of the country). 3 3 Application: The Case of Colomia Over the past 2 years, the countries of Latin America and the Cariean have made notale improvements in reducing poverty and income inequality. Income concentration, measured y the Gini coefficient, has een steadily falling in the region, going from.58 in 1996 to.52 in 211 (World Bank 213). However, in the case of Colomia, the Gini coefficient remained practically stagnant from 22 to 212, and declined only marginally during the second part of the decade. Moreover, in 212 Colomia s Gini coefficient remained higher than the regional average, placing Colomia among the three most unequal countries in LAC and one of the most unequal in the world (World Bank 213). Furthermore, these numers do not take into account the potential ias associated with the top of the income distriution, which, if properly accounted for, could result in a higher Gini coefficient. In recent years, an increasing numer of governments have een granting pulic access to administrative records and other information. The use of administrative records as a statistical tool is a recent trend that increases transparency and makes availale to citizens, 3 The control for population is computed as the total numer of individuals over a certain age (here, 2 years) in the country. The threshold depends on the age at which young people enter the laor force. See Atkinson (27). 1

14 analysts, and policymakers a greater wealth of information. Tax records contain extensive information on firms and individuals that is often not captured y traditional data collection mechanisms such as household surveys, and, as such, they can provide valuale input to statistical systems. However, many Latin American countries have yet to make pulic their income tax data. Colomia is the first country to share the disaggregated micro-data from personal income tax records. This section applies the methodology descried aove to compute the Gini coefficient for Colomia in 21 using data from oth household surveys and administrative tax records. 3.1 Data Around 1.4 percent 4 of Colomia s working population pays income taxes. While 1.1 million individuals (out of a population of 4.5 million) filed an annual tax return in 21, the rest of the population (3.4 million) paid taxes through withholding. Unlike other countries like the United States, Colomia s income tax system does not allow for joint returns for married couples. This allows us to compare incomes across data sources household survey and tax records using the same unit of analysis (individuals). Colomians file their income tax returns differently depending on whether they are small or large taxpayers. This analysis considers oth types of taxpayers: those who are not required to keep accounting ooks (small taxpayers) and those who are required to do so (large taxpayers). The small taxpayer data (Form 21) is composed of a alanced microdata panel and taulations made y the tax agency in Colomia (Dirección de Impuestos y Aduanas Nacionales, DIAN). In the case of large taxpayers (Form 11), the data set is only composed of a alanced micro-data panel. 5 The data provides annual information aout laor income, capital gains, other incomes, deductions, exemptions, and taxes paid. The Great Integrated Household Survey (GEIH) collects information aout laor force conditions, socio-demographic characteristics, and different sources of income. The ottom 4 Dirección de Impuestos y Aduanas Nacionales (DIAN), Exposición de Motivos. 5 Since we are only considering a alanced panel in which individuals are oserved if they are present for all the years, the results could e potentially iased. 11

15 of the income distriution is well represented in the GEIH, which collects data monthly and provides information at the national, uran, and rural levels as well as at the departmental level. In order to compare annual values from the administrative data with those from the household survey, we multiply y 14 the total individual monthly income (12 months plus a onus equivalent to 2 months pay). 3.2 Results In general, all individuals whose incomes exceed a certain threshold appear as a tax unit in the administrative data. 6 Individuals with incomes elow this threshold are not captured y the tax records, ut they are represented in the household survey. In 21, this threshold corresponded to 81 million Colomian pesos per year. As explained aove, this income level is used to determine the parameter of the formulas presented in Propositions 1 and 2: the ottom conditional distriution in the propositions will e otained from the household survey y focusing on all individuals with incomes elow, and the top conditional distriution will use all availale tax records (since the threshold is chosen to e the minimum income level that appears in the tax record). Figure 1 depicts the distriution of total individual incomes in the household survey. 7 Since tax records are ased on individual returns, our estimation only considers the total individual income data collected through household surveys. If more than one person per household files a tax return, they will appear separately, each with their own individual income. We define as a control for population all individuals age 2 and aove, since few individuals under 2 years of age 8 contriute income tax revenue; excluding them from the denominator does not significantly affect the results (Atkinson 27). The red vertical line in Figure 1 indicates the minimum total individual income needed 6 A caveat is worth mentioning here: the level of tax evasion and its changes over time can affect the results otained for the Gini coefficient. 7 We calculate the total income distriution considering all personal incomes: laor income, transfers, remittances, and capital gains, among others. We also include pension claims, since in Colomia pension payments are considered laor income. 8 The control for population uses population projections data for 21. The total population is 28,14,

16 # (a) Household#surveys,#21# 8.e-8 # 6.e-8 F()# Density 4.e-8 2.e-8 5.e+7 1.e+8 1.5e+8 2.e+8 Figure 1: Total Individual Income Distriution for Colomia, Household Survey. Source: # author s calculations Source:#author s#calculations#using#the#great#integrated#household#survey#(geih),#21.# using the Survey Note: The analysis considers Note:#The#analysis#considers#only#positive#total#individual#incomes#and#the#population#aove#the#age#of#2.## only total individual incomes and the population aove the age of red ar indicates the legal threshold for mandatory tax filing,. # Total Income # to e part of the tax records data, which, as discussed aove, is the threshold for the # methodology presented in this paper. The proportion of individuals who are not required ()#Tax#records,#21# to file a tax return is F (). The figure illustrates the low representation of the richest individuals in the household survey. are captured y the tax records, as shown in Figure 2. 9 Incomes aove the threshold (81 million pesos) Income is defined to include all sources reported y the individual in the tax return: wages and other laor payments; fees, commissions, and services; interest and financial income; other income (rents) minus 1/6 of the cost of deductions. 1 This definition of income is used in order to make it comparale with the income reported in the household survey, using the same adjustments as Alvaredo and Londoño (213). Tale 1 shows the estimates of the Gini coefficient using the methodology descried 9 Figure 2 is truncated at the highest income levels for readaility. 1 Using 1/6 of the cost of deductions is ad hoc. The tax file does not provide any information aout which deductions are included in this category. For more details, see Alvaredo and Londoño (213). 13 # 14#

17 1.5e-8 # 1.e-8 Density 1U#F()# 5.e-9 2.e+8 4.e+8 6.e+8 8.e+8 1.e+9 Total Income Figure 2: Total Individual Income Distriution for Colomia, Tax Records. Source: # author s calculations Source:#author s#calculations#using#income#tax#data#(dian),#21.#the#analysis#considers#only#positive#tota using income tax Note: analysis considers only total individual#incomes#and#the#population##aove#the#age#of#2.## incomes and aove age of The red ar indicates the legal threshold! for mandatory tax filing,. Tale#1#shows#the#estimates#of#the#Gini#coefficient#using#the#methodology#descried#in#Section#2 in Section 2, setting the threshold to various levels. The old row corresponds to the setting#the#threshold##to#various#levels.#the#old#row#corresponds#to#the#optimal##(the#minimum optimal (the minimum income that mandates filing of the tax record). The first column income#that#mandates#filing#of#the#tax#record).#the#first#column#reports#the#level#of#,#the#second reports column#the#estimated#fraction#of#the#population#with#incomes#elow#,#or#f().#columns#3#and#4 the level of, the second column the estimated fraction of the population with incomesreport# elowthe#, or Gini# F (). coefficient# Columnsestimated# 3 and 4 report using# the the# Gini conditional# coefficient distriution# estimated of# using incomes# the elow# # and conditional aove#,#respectively.#finally,#column#5#reports#the#estimate#of#the#gini#coefficient#that#comines distriution of incomes elow and, column 5 reportsthe# two# estimate numers# of the as# Gini descried# coefficient in# Proposition# that comines 2.# The# the two resulting# numers preferred# as descried estimate# of# the# Gin in Proposition coefficient# 2. is# The.5978.# resulting The# preferred estimate# estimate varies# minimally# of the Gini with# coefficient the# choice# is of# # around# The the# optima estimate threshold# varies minimally of# 81# million# with pesos.# the choice However,# of around choosing# thea# optimal higher# threshold for# example# of 81 million focusing# on# the# top# 1 pesos. percent# However, of# choosing the# income# a higher distriution# for(=16# examplemillion# focusing pesos) does# on the top have# 1 percent a# nontrivial# of the impact# on# the incomegini#coefficient,#y#aout#.2#points#(see#last#row#of#tale#1).#considering#that#the#average#annua distriution ( = 16 million pesos) does have a nontrivial impact on the coefficient, reduction# y aout of# the#.2 Gini# points coefficient# (see last rowlatin# of Tale American# 1). Considering countries# over# thatthe# last# average 1# years# was#.51 annual percentage#points,#the#differences#in#the#estimation#are#not#trivial.#! reduction of the Gini coefficient Latin American countries over the last 1 years was.51! percentage points, the differences in the estimates are not trivial. Tale! 1:! Gini! calculations! for! the! conditional! distriution! of! income! aove! and! elow! a threshold! 14

18 F () G[F 1 ] G[F 2 ] G[F ] 79,, ,, ,31, ,, ,, ,7, Tale 1: Gini coefficient estimates using different thresholds. Column (1) reports the chosen coefficient. Column (2) reports the estimated F (). Column (3) reports G[F 1 ], the Gini coefficient computed using the incomes elow (using the Household Survey), Column (4) reports G[F 2 ], the Gini coefficient computed using the incomes aove (using the Tax Records), and Column (5) reports G[F ], the overall estimated Gini coefficient. The old line corresponds to the optimal choice of. An even more striking difference can e seen when comparing these numers to the ones otained y Alvaredo and Londoño (213) for Colomia applying the methodology of Alvaredo (211). As mentioned earlier, in this paper we uild on his methodology and propose an improvement ased on the optimal choice of the income threshold used to comine the household surveys with the tax records. In particular, the main differences etween their methodology and the one presented in this paper are: (1) the use of a threshold of 1 percent to comine the two data sets, instead of choosing an optimal threshold for comining the two; (2) the use of an approximation to comine the Gini coefficient of the ottom of the income distriution otained from the household survey with the share of income of the top 1 percent of the population; and (3) the calculation of the Gini coefficient for the ottom 99 percent of the income distriution using the ottom 99th percentile of the income distriution otained from the household survey (ignoring the underrepresentation at the top in the household survey). Applying the Alvaredo and Londoño methodology to the Colomian case, the resulting Gini coefficient would e.676,.98 percentage points higher than the estimate otained using the methodology presented 15

19 in this paper (the old row of Tale 1). This shows that choosing an optimal threshold to comine the two datasets and explicitly considering the underreporting of incomes in the household survey when computing the Gini coefficients for conditional distriutions can have dramatic effects on the measured coefficient Conclusion This paper proposes a new methodology to optimally measure inequality y comining household survey and tax records data. The motivation stems from the fact that household surveys poorly represent the top of the income distriution, while tax records cover the top ut not the ottom of the distriution. The key innovation of the proposed methodology is the choice of an income threshold used to comine the two data sources; the paper shows that should not e chosen aritrarily, ut should e chosen optimally to minimize the distortions from household survey underreporting at the top of the income distriution. In particular, we discuss why the optimal income threshold corresponds to the minimum income level required to file a tax return (i.e., the lowest income captured in the tax records). After presenting the methodology, we apply it to the case of Colomia. We find that the Gini coefficient for Colomia in 21 is.5978 when computed using an optimal income threshold of 81 million pesos, which is significantly different from the one that would e otained if using an (aritrary) threshold corresponding to the top 1 percent of the income distriution (.596). The difference of.2 points is not trivial, particularly when we consider that over the last 1 years the average annual reduction of the Gini coefficient in Latin America was.51 percentage points. The methodology presented in this paper uilds on and improves upon the methodologies proposed y Atkinson (27) and Alvaredo (211). 11 When applying the methodology presented in Alvaredo (211) and Alvaredo and Londoño (213), instead of estimating the Gini coefficient for the ottom 99 percent of the population using the ottom 99 percent of the household survey s distriution, an alternative could e to simply use the Gini coefficient of the full distriution of the household survey as an estimator of the Gini coefficient for the ottom 99 percent of the true income distriution. This would e correct under the assumption that the household survey underreports incomes exactly aove the 99th percentile. The Gini coefficient computed under this assumption (which generally will not e true) would e

20 In addition to improving the measurement of inequality, the new proposed methodology is useful for assessing the distriutional effects of fiscal policies on the income distriution, since it also allows to recover the full unconditional distriution of income y comining the conditional distriutions oserved in the household surveys and in the tax records. 17

21 References 1. Alvaredo, Facundo A note on the relationship etween top income shares and the Gini coefficient. Economics Letters 11 (3): Alvaredo, Facundo, and Juliana Londoño Velez High Incomes and Personal Taxation in a Developing Economy: Colomia Commitment to Equity Working Paper No. 12, Tulane University, New Orleans, LA. 3. Atkinson, A.B., 27. Measuring Top Incomes: Methodological Issues. In: Atkinson, A., Piketty, T. (Eds.), Top Incomes over the Twentieth Century: A Contrast Between Continental European and English-Speaking Countries. Oxford University Press, Oxford. 4. Burkhauser, Richard V., Kenneth A. Couch, Andrew Houtenville, and Ludmila Rova. 24. Income Inequality in the 199s: Re-Forging a Lost Relationship?, Working Papers 24 11, University of Connecticut, Department of Economics. 5. Burkhauser, Richard V., Shuaizhang Feng, Stephen P. Jenkins, and Jeff Larrimore. 29. Recent Trends in Top Income Shares in the USA: Reconciling Estimates from March CPS and IRS Tax Return Data. NBER Working Papers 1532, National Bureau of Economic Research, Inc Recent Trends in Top Income Shares in the United States: Reconciling Estimates from March CPS and IRS Tax Return Data. The Review of Economics and Statistics 94 (2): Cowell, Frank A., and Maria-Pia Victoria-Feser. 2. Distriutional Analysis: A Roust Approach. In Putting Economics to Work, eds. Anthony Atkinson, Howard Glennerster, and Nicholas Stern, pp London: STICERD, London School of Economics Distriution-free inference for welfare indices under complete an incomplete information. Journal of Economic Inequality 1: Dagum, Camilo A New Approach to the Decomposition of the Gini Income Inequality Ratio. Empiral Economics 22:

22 1. Dirección de Impuestos y Aduanas Nacionales (DIAN) Exposición de motivos al proyecto de Ley. Ministerio de Hacienda y Crédito Púlico, Government of Colomia, Bogotá. 11. Estatuto Triutario Nacional Government of Colomia, Bogotá. 12. Gasparini, Leonardo, Mariana Marchionni, and Walter Sosa Escudero. 2. La distriución del ingreso en Argentina y en la provincia de Buenos Aires. Ministerio de Economía de la Provincia de Buenos Aires, Repúlica Argentina. 13. Székely, Miguel, and Marianne Hilgert What s Behind the Inequality We Measure: An Investigation Using Latin American Data. Working Paper No. 49, Inter-American Development Bank, Washington, DC. 14. World Bank Shifting Gears to Accelerate Shared Prosperity in Latin America and the Cariean. Document 7857, World Bank, Washington, DC. 19

23 Appendix Proof of Proposition 2 Start y writing: G[F ] = 1 F (y)(1 F (y))dy = 1 Sustituting the expressions aove: + 1 Now note that F (y)(1 F (y))dy + 1 G[F ] = 1 F (y)(1 F (y))dy F ()F 1 (y)(1 F ()F 1 (y))dy [F () + (1 F ())F 2 (y)](1 F () (1 F ())F 2 (y)])dy F 1 (y)f ()(1 F 1 (y)f ()) = F 1 (y)f () F 1 (y) 2 F () 2 = = F () 2 [F 1 (y) F 1 (y) 2 ] + F 1 (y)[f () F () 2 ] Similarly, we otain [F () + (1 F ())F 2 (y)][1 F () (1 F ())F 2 (y)] = F () + (1 F ())F 2 (y) [F () + (1 F ())F 2 (y)] 2 = F () + (1 F ())F 2 (y) [F () 2 + (1 F ()) 2 F 2 (y) 2 + 2F ()(1 F ())F 2 (y)] = F () + (1 F ())F 2 (y) F () 2 (1 F ()) 2 F 2 (y) 2 2F ()(1 F ())F 2 (y) = F () + (1 F ())F 2 (y) F () 2 + (1 F ()) 2 [ F 2 (y) F 2 (y) 2] (1 F ()) 2 F 2 (y) 2F ()(1 F ())F 2 (y) 2

24 = F () + [ (1 F ()) (1 F ()) 2 2F ()(1 F ()) ] F 2 (y) F () 2 + (1 F ()) 2 [ F 2 (y) F 2 (y) 2] = F () + [ 1 F () (1 + F () 2 2F ()) 2F () + 2F () 2] F 2 (y) F () 2 + (1 F ()) 2 [ F 2 (y) F 2 (y) 2] = F () + [ F () + F () 2] F 2 (y) F () 2 + (1 F ()) 2 [ F 2 (y) F 2 (y) 2] = [F () F () 2 ] F ()(1 F ())F 2 (y) + (1 F ()) 2 [ F 2 (y) F 2 (y) 2] Sustituting: G[F ] = { F () 2 [F 1 (y) F 1 (y) 2 ] + F 1 (y)[f () F () 2 ] } dy { [F () F () 2 ] F ()(1 F ())F 2 (y) + (1 F ()) 2 [ F 2 (y) F 2 (y) 2]} dy G[F ] = F () + 1 F () { F ()[F1 (y) F 1 (y) 2 ] + F 1 (y)[1 F ()] } dy { F () F ()F2 (y) + (1 F ()) [ F 2 (y) F 2 (y) 2]} dy Now note that y Fuini s theorem, since income is all nonnegative, = E[Y ] = (1 F (y))dy Also (1 F (y))dy = (1 F (y))dy + (1 F (y))dy = = = 1 F ()F 1 (y)dy + 1 F ()F 1 (y)dy + 1dy F () F 1 (y)dy + 1 [F () + (1 F ())F 2 (y)]dy (1 F ()) (1 F ())F 2 (y)dy (1 F ())(1 F 2 (y))dy 21

25 = 1dy F () = (1 F ()) + F () So: F 1 (y)dy + F () = (1 F ()) + F () (1 F 1 (y))dy + 1dy F () (1 F 1 (y))dy + 1dy + (1 F ())(1 F 2 (y))dy (1 F ())(1 F 2 (y))dy (1 F ())(1 F 2 (y))dy + (1 F ()) (1 F ()) = F () 1 + (1 F ()) 2 Now we are going to focus only on one part of G[F ], that does not involve F (y) 2 terms: + F () G[F ] = F () + 1 F () = F () The first part (first line) is: { F ()[F1 (y) F 1 (y) 2 ] + F 1 (y)[1 F ()] } dy { F () F ()F2 (y) + (1 F ()) [ F 2 (y) F 2 (y) 2]} dy {F 1 (y)[1 F ()]} dy + 1 F () { F ()[F1 (y) F 1 (y) 2 ] } dy + 1 F () F () {F 1 (y)[1 F ()]} dy + 1 F () {F () F ()F 2 (y)} dy { (1 F ()) [ F2 (y) F 2 (y) 2]} dy {F () F ()F 2 (y)} dy = (1 F ())F () [ F 1 (y)dy + {1 F 2 (y)} dy] Now rememer that the the second distriution F 2 has support (, ), and is elow, so that the mean income of the upper distriution can e computed as (these are conditional means): 2 = + and the average income of the first distriution is: 1 = (1 F 1 (y))dy = dy {1 F 2 (y)} dy F 1 (y)dy = F 1 (y)dy 22

26 so F 1 (y)dy = 1 {1 F 2 (y)} dy = 2 F () {F 1 (y)[1 F ()]} dy + 1 F () {F () F ()F 2 (y)} dy = So we can write: G[F ] = F () = (1 F ())F ()[ 2 1 ] { F ()[F1 (y) F 1 (y) 2 ] } dy + 1 F () Now we can look at the first part: F () = F ()2 +(1 F ())F ()[ 2 1 ] { F ()[F1 (y) F 1 (y) 2 ] } dy + 1 F () 1 1 So to conclude we have: { [F1 (y) F 1 (y) 2 ] } dy + [1 F ()]2 { (1 F ()) [ F2 (y) F 2 (y) 2]} dy { (1 F ()) [ F2 (y) F 2 (y) 2]} dy 2 2 = F () 2 1 G[F 1] + [1 F ()] 2 2 G[F 2] {[ F2 (y) F 2 (y) 2]} dy G[F ] = F () 2 1 G[F 1] + [1 F ()] 2 2 G[F 2] + (1 F ())F ()[ 2 1 ] 23